17.7: Additional Remarks
- Page ID
- 5833
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The units erg \(\text{G}^{-1}\) or \(\text{J T}^{-1}\) are frequently encountered for magnetic moment. These may be dimensionally correct, although ergs and joules (units of energy) are not quite the same things as \(\text{dyn cm}\) or \(\text{N m}\) as units of torque. It could be argued that magnetic moment could be defined from the expression \(-\boldsymbol{p.B}\) for the potential energy of a magnet in a magnetic field. But the correct expression is actually constant \(-\boldsymbol{p.B}\), the constant being zero only if you specify that the energy is taken to be zero when the magnetic moment and field vectors are perpendicular to each other. This seems merely to add yet further complications to what should be, but unfortunately is not, a concept of the utmost simplicity.
Nevertheless the use of ergs or joules rather than \(\text{dyn cm}\) or \(\text{N m}\) is not uncommon, and nuclear and particle physicists commonly convert joules to \(\text{MeV}\). Magnetic moments of atomic nuclei are commonly quoted in nuclear magnetons, where a nuclear magneton is \(e\hbar / (2m_p)\) and has the value \(3.15 \times 10^{-4} \ \text{MeV T}^{-1}\). While one is never likely to want to express the magnetic moment of the planet Uranus in nuclear magnetons, it is sobering to attempt to do so, given that the magnetic moment of Uranus is quoted as \(0.42 \ \text{Oe km}^{-1}\). While on the subject of Uranus, I have seen it stated that the magnetic quadrupole of Uranus is or the same order of magnitude as its magnetic dipole moment – though, since these are dimensionally dissimilar quantities, such a statement conveys no meaning.
Another exercise to illustrate the points I have been trying to make is as follows. From four published papers I find the following. The magnetic moment of Mercury is \(1.2 \times 10^{19} \ \text{A m}^2\) in one paper, and \(300 \ \text{nT} \ {\text{R}_M}^3\) in another. The magnetic moment of Uranus is \(4.2 \times 10^{12} \ \text{Oe km}^3\) in one paper, and \(0.23 \ \text{G} \ {\text{R}_U}^3\) in another. The radii of Mercury and Uranus are, respectively, \(2.49 \times 10^6 \ \text{m}\) and \(2.63 \times 10^7 \ \text{m}\). Calculate the ratio of the magnetic moment of Uranus to that of Mercury. If you are by now completely confused, you are not alone.